Rigid Body Research Articles

A Generalized 3D-DDM for both Fracture-Type and Asymmetric Closed-Contour Problems: Modified Fully Analytical Approach and Applications

The displacement discontinuity method (DDM) has been widely used for large-scale engineering problems, e.g., hydraulic fracturing related to metal mining and hydrocarbon extraction. However, there is a lack of detailed explanation for the implementation of 3D-DDM in closed-contour boundary problem, and applications to address asymmetric closed-contour boundary problem remains limited. In this paper, we have theoretically derived and elaborated the numerical procedure, identified the caveats of DDM in solving fracture-type and closed-contour boundary problems. Subsequently, we modified the analytical integral and differential expressions for the basic solution on 3D-DDM based on triangular elements and provided a more general suite of expressions. Additionally, implementation details including element traversal criteria and precautions against rigid body movement are introduced to address the asymmetric closed-contour boundary problem. The accuracy of the generalized 3D-DDM were verified through simulations involving a penny-shaped fracture and a spherical cavity. Finally, the generalized 3D-DDM was employed to elucidate the stress redistribution and interference with respect to the tunnel excavation. This study demonstrates that: (1) the modified analytical integral and differential expressions combined with the element traversal criterion and precautions against rigid body movement provide a detailed explanation of how 3D-DDM can be applied to asymmetric closed-contour boundary problem; (2) during tunnel excavation, the arch flanks and floor would suffer from the biggest damage risk; and (3) the interaction effect between adjacent tunnels plays a role when the tunnel spacing is less than eight times the radius of the arch.

Dynamical Modeling and Dynamic Characteristics Analysis of a Coaxial Dual-Rotor System

The dual-rotor structure is the primary excitation source of aero-engine vibration. It is essential to study the dynamical model and analyse the dynamic characteristics such as critical speed and the unbalanced response for rotor system dynamics. The coaxial dual-rotor support scheme of an aero-engine was introduced in the previous work, and the physical model with a high-speed flexible inner rotor of a sizeable length-diameter ratio has been designed. Then a finite element dynamic model based on the Timoshenko beam elements and rigid body kinematics of the dual-rotor system is modelled, with the Newmark-β method and Newton-Raphson method used for the numerical calculation to study the dynamic characteristics of the system. Three different simulation models, including beam-based FE (1D finite element) model, solid-based FE (3D finite element) model, and TM (transfer matrix) model, were designed to study the characteristics of mode and the critical speed characteristic of the dual-rotor system. The unbalanced response of the dual-rotor system was analysed to study the influence of mass unbalance on the rotor system. The effect of different disc unbalance phases and different speed ratios on the dynamic characteristics of the dual-rotor system were investigated in detail. The experimental result shows that the beam-based FE model is effective and suitable for studying the dual-rotor system. Conflict of Interest Statement The authors declare no conflicts of interest.

A centroid measurement method of large components for active compliant assembly

In the active compliant assembly system of large components, the centroid parameter is an important factor affecting the dynamic model of the attitude adjustment mechanism. This paper proposes a novel automatic centroid measurement method. Firstly, the contact force calculation method between large components during active compliant assembly was described, and the impact of the centroid of large components on the accuracy of the contact force calculation was analyzed. Secondly, based on the principles of torque balance and rigid body rotation, a novel centroid calculation model considering lateral forces is established, and RANSAC is used to optimize the centroid solution. Then, numerical simulation methods were used to analyze the impact of random errors and weighing strategies on the centroid measurement accuracy, and suggestions were given to improve the centroid measurement accuracy. Finally, a compliant assembly system for large component was built in the laboratory, and centroid measurement and accuracy verification experiments were conducted. The experimental results show that the proposed method can significantly improve the accuracy of centroid measurement and can effectively reduce the contact force calculation error during the compliant assembly process of large components.

Analytical Solutions of the 3-DOF Gyroscope Model

The motion of a rigid body (a gyroscope) is one of the key issues in classical mechanics. It remains a significant challenge, as evidenced by its extensive practical implementations in various scientific disciplines and engineering operations. It is important to obtain analytical solutions, as they provide solutions that depend directly on the system’s parameters, which can be definitively interpreted. The coupling of numerical and analytical solutions allows for a more precise representation of the real phenomenon. The main objective of the article was to formulate analytical solutions for the motion of a Cardan suspension gyroscope subjected to controlling torque moments. Analytical solutions for the proposed mathematical model were developed using the Laplace transform and Green’s function. Subsequently, they were validated by numerical tests. The obtained analytical solutions are universally applicable, regardless of the type of controlling moments.

Quasi-steady aerodynamic modeling and dynamic stability of mosquito-inspired flapping wing pico aerial vehicle

Recent exploration in insect-inspired robotics has generated considerable interest. Among insects navigating at low Reynolds numbers, mosquitoes exhibit distinct flight characteristics, including higher wingbeat frequencies, reduced stroke amplitudes, and slender wings. This leads to unique aerodynamic traits such as trailing edge vortices via wake capture, diminished reliance on leading vortices, and rotational drag. This paper shows the energetic analysis of a mosquito-inspired flapping-wing Pico aerial vehicle during hovering, contributing insights to its future design and fabrication. The investigation relies on kinematic and quasi-steady aerodynamic modeling of a symmetric flapping-wing model with a wingspan of approximately 26 mm, considering translational, rotational, and wake capture force components. The control strategy adapts existing bird flapping wing approaches to accommodate insect wing kinematics and aerodynamic features. Flight controller design is grounded in understanding the impact of kinematics on wing forces. Additionally, a thorough analysis of the dynamic stability of the mosquito-inspired PAV model is conducted, revealing favorable controller response and maneuverability at a small scale. The modified model, incorporating rigid body dynamics and non-averaged aerodynamics, exhibits weak stability without a controller or sufficient power density. However, the controller effectively stabilizes the PAV model, addressing attitude and maneuverability. These preliminary findings offer valuable insights for the mechanical design, aerodynamics, and fabrication of RoboMos, an insect-inspired flapping wing pico aerial vehicle developed at UPM Malaysia.

Initiation of decohesion between a flat punch and a thin bonded incompressible layer

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson’s equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact.

Mechanism and Application of Attitude and Orbit Coupling Dynamics for Spacecraft Proximity Relative Motion

This paper analyzes the root causes of attitude-orbit coupling effects of spacecraft proximity relative motion in space precision collaborative tasks from three aspects: mathematical representation, physical definition, and engineering applications. At first, taking mathematical representation as the context, spacecraft proximity relative motion representations such as particle relative dynamic model, extended particle relative dynamic model, and dual-spiral-based relative dynamic model are investigated in detail. On this basis, the mechanism of attitude-orbit coupling effects originating from different mathematical representations is further investigated. Second, spiral theory–based attitude-orbit coupling relative dynamics is developed. The innovation of this work is extending the dual number representation from rigid body to flexible body, which makes it possible to describe the proximity relative motion between two rigid-flexible coupling spacecraft. Third, the application value of attitude-orbit coupling relative dynamic model in precision collaborative mission such as precision formation, rendezvous and docking, space manipulation, and on-orbit assembly is provided. Finally, simulation results verify the engineering significance of the attitude-orbit coupling relative dynamic model.

Modelling Rigid Body Potential of Small Celestial Bodies for Analyzing Orbit–Attitude Coupled Motions of Spacecraft

The present study aims to propose a general framework of modeling rigid body potentials (RBPs) suitable for analyzing the orbit–attitude coupled motion of a spacecraft (S/C) near small celestial bodies, regardless of gravity estimation models. Here, ‘rigid body potential’ refers to the potential of a small celestial body integrated across the finite volume of an S/C, assuming that the mass of the S/C has no influence on the motion of the small celestial body. First proposed is a comprehensive formulation for modeling the RBP including its associated force, torque, and Hessian matrix, which is then applied to three gravity estimation models. The Hessian of potential plays a crucial role in calculating the RBP. This study assesses the RBP via numerical simulations for the purpose of determining proper gravity estimation models and seeking modeling conditions. The gravity estimation models and the associated RBP are tested for eight small celestial bodies. In this study, we utilize distance units (DUs) instead of SI units, where the DU is defined as the mean radius of the given small celestial body. For a given specific distance in Dus, the relative error of the gravity estimation model at this distance has a similar value regardless of the small celestial body. However, the difference value between the potential and RBP depends on the DU; in other words, it depends on the size of the small celestial body. This implies that accurate gravity estimation models are imperative for conducting RBP analysis. The overall results can help develop a propagation system for orbit–attitude coupled motions of an S/C in the vicinity of small celestial bodies.

Straw movement and flow field in a crushing device based on CFD-DEM coupling with flexible hollow straw model

To accurately predict and evaluate the performance of the crushing device of no-till planter, it is strongly necessary to establish an accurate interaction model of the straw-crushing device coupling system. The traditional model has low prediction accuracy and cannot simulate the fracture characteristics of the straw during the operation of the crushing device due to the straw being usually regarded as a rigid body or a combination of rigid ball joints. In this paper, a breakable and flexible hollow straw model is first proposed and a CFD-DEM coupling model of the straw group-crushing device considering the effect of gas-solid two phase is also established. Afterwards, the corresponding verification experiments are conducted and the effect of working and structural parameters on the straw-throwing weight and crushing length is also analysed. It is demonstrated that the relative error between the simulation results and the experimental results for the straw crushing length is 7.7%, and the relative error for the straw-throwing weight is 7.1%, thereby verifying the correctness of the CFD-DEM model. Finally, the response surface method is also used to determine the optimal working and structural parameters of the crushing device and corresponding field tests verification is also conducted. Results indicate that the optimal parameter combination is as: the rotational speed of the crushing spindle is 2400 rpm, the inlet clearance is 30 mm and the starting point angle is 45°. Moreover, the crushing length after optimisation is 6.6% smaller than before optimisation, and the straw-throwing weight is increased by 11.5%.

Output feedback control for hypersonic flight vehicles via distributed high-gain observer

During the hypersonic flight process of rigid hypersonic flight vehicles (HFVs), the flight-path angle and angle of attack are usually small and sensitive to flight condition variation, the accurate measurements are difficult, subsequently the formulation of states observers is an urgent issue. A distributed high-gain observer is proposed to reconstruct the admissible system state dynamics based on limited measured output. First, a distributed observer is proposed for the cascade strict-feedback system to ensure the global convergence of the state estimation errors between the estimated state in each local observer and system states, for the prescribed system initial values. Second, the distributed high-gain observer-based nonlinear control is investigated on the longitudinal dynamics of HFVs with partial measurable states. The rigid body system dynamic is divided into the altitude subsystem and the velocity subsystem. In the altitude subsystem, the proposed distributed high-gain observer is formulated to exact estimate the flight-path angle and angle of attack, and then the back-stepping design is proposed for constructing the controllers. The nonlinear dynamic inversion controller is introduced to accomplish the velocity tracking. Finally, simulation examples are served to verify that the proposed output feedback control possesses the superior properties of high tracking performance, fast convergence of state estimation error and easy-implementation.

Real-Time Seamless Multi-Projector Displays on Deformable Surfaces.

Prior works on multi-projector displays have focused primarily on static rigid objects, some focusing on dynamic rigid objects. However, works on projection based displays on deformable dynamic objects have focused only on small scale single projector displays. Tracking a deformable dynamic surface and updating projections precisely in real time on it is a significantly challenging task, even for a single projector system. In this paper, we present the first end-to-end solution for achieving a real-time, seamless display on deformable surfaces using mutliple unsychronized projectors without requiring any prior knowledge of the surface or device parameters. The system first accurately calibrates multiple RGB-D cameras and projectors using the deformable display surface itself, and then using those calibrated devices, tracks the continuous changes in the surface shape. Based on the deformation and projector calibration, the system warps and blends the image content in real-time to create a seamless display on a surface that continuously changes shape. Using multiple projectors and RGB-D cameras, we provide the much desired aspect of scale to the displays on deformable surfaces. Most prior dynamic multi-projector systems assume rigid objects and depend critically on the constancy of surface normals and non-existence of local shape deformations. These assumptions break in deformable surfaces making prior techniques inapplicable. Point-based correspondences become inadequate for calibration, exacerbated with no synchronization between the projectors. A few works address non-rigid objects with several restrictions like targeting semi-deformable surfaces (e.g. human face), or using single coaxial (optically aligned) projector-camera pairs, or temporally synchronized cameras. We break loose from such restrictions and handle multiple projector systems for dynamic deformable fabric-like objects using temporally unsynchronized devices. We devise novel methods using ray and plane-based constraints imposed by the pinhole camera model to address these issues and design new blending methods dependent on 3D distances suitable for deformable surfaces. Finally, unlike all prior work with rigid dynamic surfaces that use a single RGB-D camera, we devise a method that involve all RGB-D cameras for tracking since the surface is not seen completely by a single camera. These methods enable a seamless display at scale in the presence of continuous movements and deformations. This work has tremendous applications on mobile and expeditionary systems where environmentals (e.g. wind, vibrations, suction) cannot be avoided. One can create large displays on tent walls in remote, austere military or emergency operations in minutes to support large scale command and control, mission rehearsal or training operations. It can be used to create displays on mobile and inflatable objects for tradeshows/events and touring edutainment applications.

High-Fidelity Flexible Multibody Model Considering Torsional Deformation for Nonequatorial Space Elevator

Space elevators are a promising technology for future space transportation. The primary component of a space elevator is a tether deployed from a geostationary orbit towards the Earth and into deep space. Payloads can be transported using a climber moving along the tether. However, obtaining a deeper comprehension of the precise dynamics of a space elevator is challenging. The tether could have various deformations such as elongation, bending, and torsion owing to its flexible nature and the low level of vibration damping in space. Therefore, understanding those effects on the tether is essential for the design concept of a space elevator. Reproducing the behavior of space tether systems on the ground is challenging, which is primarily due to the scale of the system and the difficulty of accurately replicating a space environment. Thus, high-fidelity numerical analysis methods are required to analyze the dynamic response of space elevators precisely. Previously proposed analysis models of space tether systems with climbers have focused on elongation, bending, and rigid body motion of the tether, and the effect of torsional deformation has been neglected. However, a tether can be twisted easily owing to its low torsional stiffness. This study established high-fidelity numerical simulation models for space elevators considering large deformations including the torsion of a tether. This study developed a flexible multibody model with a moving climber using the absolute nodal coordinate formulation with 14 degrees of freedom. Further, the gravitational perturbation caused by the non-sphericity of the Earth was considered. Using the constructed model, the motion and deformation of nonequatorial space elevators caused by the climber motion and the disturbance in the space environment were investigated. The obtained results indicate that gravitational perturbation can induce torsional deformation, particularly at higher anchor latitudes.

Quantitative analysis of deformation performance for skew slab based on deformation energy decomposition method of triangular prism element

In order to propose a simpler and more effective method to quantitatively analyze the macroscopic basic deformation properties such as tension, bending and shear of the skew slabs, and provide theoretical basis and technical support for targeted design and reinforcement. Based on mathematical orthogonality, mechanical equilibrium and physical parameters, this paper proposes a deformation energy decomposition method suitable for 3D 6-node arbitrary triangular prism elements. By applying this method to the finite element model discretized by triangular prism element, the quantitative information of 6 kinds of rigid body displacement and 12 kinds of deformation energy of skew slab can be obtained. The quantitative and visual analysis of the basic deformation performance of the skew slab is carried out by using the method in this paper. Then the targeted structural design and reinforcement design of the skew slab can be guided according to the principle of matching design with deformation. Compared with the existing deformation decomposition methods based on square elements, which are only applicable to 2D or quasi-3D regular structures, the deformation energy decomposition method based on arbitrary triangular prism elements in this paper can be used to analyze the deformation performance of 3D irregular structures. The influence of skew angle and span-width ratio on the deformation performance of skew slab is analyzed by deformation energy decomposition method. The results show that when the skew angle is greater than 25°, the proportion of ductile deformation energy such as tension and bending of skew slab decreases rapidly. In this case, the shear design of the skew slab should be focused on. When the angle is greater than 45°, the span-width ratio should be appropriately reduced to improve the ductility of the skew slab, and the torsion design needs to be focused on.

ANALYSIS OF ASYMMETRIC PLASTIC BENDING DEFORMATION OF WELDED TUBE SHEETS AND DERIVATION OF A DAMAGE LAW FOR J SHAPE-C SHAPE -O SHAPE FORMING

The J shape-C shape -O shape (JCO) forming method of sheets for large-diameter longitudinally welded pipes has been widely used to manufacture oil and gas pipelines. The quality of welded pipes formed through the JCO process is directly affected by the accurate control of asymmetric bending springback and damage during the procedure. A finite element model (FEM) for JCO asymmetric bending was constructed to investigate the characteristics of asymmetric plastic bending deformation and the distribution laws of damage in JCO forming. This model was based on a nonlinear mixed hardening Lemaitre damage model coupled with the Quasi-Plastic-Elastic (QPE) model. Then, residual stress, forming shape, and damage degree of crimping, first bending, second bending, and third bending were analyzed. Moreover, the accuracy of the model was experimentally verified. Results demonstrate that, the asymmetric bending deformation process can be divided into five stages, including rigid body rotation phase, full elastic deformation phase, elastic–plastic uncladded deformation phase, elastic–plastic cladded phase, and springback phase. In the first bending, the stress centerline significantly misaligns with the mold centerline. Asymmetry is not evident in the second and third bending stages. Furthermore, the peak residual stress and peak damage can be effectively reduced by a larger punch radius, while the peak damage can be reduced by a larger die span. This study lays a foundation for optimizing the technical parameters of JCO forming. Keywords: JCO forming, asymmetric bending, finite element, QPE model, deformation characteristics

Attitude Tracking for Rigid Bodies Using Vector and Biased Gyro Measurements

Attitude Tracking for Rigid Bodies Using Vector and Biased Gyro Measurements

Development of a New Vertical Dynamic Model of a Rail Vehicle for the Analysis of Ride Comfort

The rail vehicle industry wants to produce vehicles with higher speeds, to maintain and increase its market share. However, when the speed of the vehicle increases, it may have an undesirable effect on ride comfort, in terms of ride dynamics. Recent developments towards lighter and faster vehicles make the problem of ride comfort at higher speeds increasingly important. Focusing on the behavior of flexible rather than rigid body behavior should not be neglected when designing long and light car bodies. There are several approaches to incorporate body flexibility in multibody simulations and they have some superiorities and weaknesses. In this study, an efficient and accurate vertical dynamic model for the ride comfort analysis is developed and implemented in a commercial object-oriented modeling (OOM) software Dymola (2015 FD01) which uses the open-source code Modelica. This model includes car body flexibility with the assembling of a rigid body approach. The developed model is compared to a three-dimensional vehicle model in the commercial Vampire software (Pro V5.50) at different velocities. For the vertical ride comfort analysis, the ISO 2631-1 standard was used for both the developed model and the three-dimensional model. The results are presented as acceleration history and awrms—weighted r.m.s (root mean square) of accelerations—as required by the standard. The developed model has shown its feasibility in terms of its efficiency and accuracy for the vertical ride comfort analysis. The accuracy of the model is evidenced by the fact that the car body vibration level at high speeds shows minor differences compared to the results of the Vampire, which is a validated commercial software in the area of rail vehicle dynamics. The approach involving the assembly of rigid bodies is applied for the first time for high-speed trains in dynamical modelling, with flexible car bodies for ride comfort analysis. Furthermore, it can be used for parametrical studies focusing on ride comfort, thereby offering a quite beneficial framework for addressing the challenges of ride comfort analysis in high-speed rail vehicles. Improvements for and analyses of other aspects are also possible, since the optimization and other useful libraries are readily available in Dymola/Modelica.

An ontology of 3D environment where a simulated manipulation task takes place (ENVON)

Thanks to the advent of robotics in shopfloor and warehouse environments, control rooms need to seamlessly exchange information regarding the dynamically changing 3D environment to facilitate tasks and path planning for the robots. Adding to the complexity, this type of environment is heterogeneous as it includes both free space and various types of rigid bodies (equipment, materials, humans etc.). At the same time, 3D environment-related information is also required by the virtual applications (e.g., VR techniques) for the behavioral study of CAD-based product models or simulation of CNC operations. In past research, information models for such heterogeneous 3D environments are often built without ensuring connection among different levels of abstractions required for different applications. For addressing such multiple points of view and modelling requirements for 3D objects and environments, this paper proposes an ontology model that integrates the contextual, topologic, and geometric information of both the rigid bodies and the free space. The ontology provides an evolvable knowledge model that can support simulated task-related information in general. This ontology aims to greatly improve interoperability as a path planning system (e.g., robot) and will be able to deal with different applications by simply updating the contextual semantics related to some targeted application while keeping the geometric and topological models intact by leveraging the semantic link among the models.

Integrating Computational Fluid Dynamics for Maneuverability Prediction in Dual Full Rotary Propulsion Ships: A 4-DOF Mathematical Model Approach

To predict the maneuverability of a dual full rotary propulsion ship quickly and accurately, the integrated computational fluid dynamics (CFD) and mathematical model approach is performed to simulate the ship turning and zigzag tests, which are then compared and validated against a full-scale trial carried out under actual sea conditions. Initially, the RANS equations are solved, employing the Volume of Fluid (VOF) method to capture the free water surface, while a numerical simulation of the captive model test is conducted using the rigid body motion module. Secondly, hydrodynamic derivatives for the MMG model are obtained from the CFD simulations and empirical formula. Lastly, a four-degree-of-freedom mathematical model group (MMG) maneuvering model is proposed for the dual full rotary propulsion ship, incorporating full-scale simulations of turning and zigzag tests followed by a full-scale trial for comparative validation. The results indicate that the proposed method has a high accuracy in predicting the maneuverability of dual full-rotary propulsion ships, with an average error of less than 10% from the full-scale trial data (and within 5% for the tactical diameters in particular) in spite of the influence of environmental factors such as wind and waves. It provides experience in predicting the maneuverability of a full-scale ship during the ship design stage.

Analyzing the dynamics of a charged rotating rigid body under constant torques

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler’s equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler’s angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body’s parameters have on the motion’s behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body’s motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance.

Approaches for Fast Brownian Dynamics Simulation with Constraints

Constraints can be used to eliminate quickly fluctuating degrees of freedom in dynamical systems enabling solutions that resolve the behavior of the system with fewer time steps over long time scales. In Brownian dynamics models, these constraints lead to linear equations which represent saddle point problems describing the velocity tangent to the constraint manifold and the forces resulting from imposition of the constraint on the dynamics. Simulations of these constrained systems must solve these saddle point problems, which are often ill-conditioned and require clever preconditioning in order to facilitate large scale simulation using iterative methods like GMRES. In this work, we show that the projected conjugate gradient (PrCG) method, which applies conjugate gradients projected onto the tangent space of the constraint manifold, has certain advantages compared to preconditioned iterative methods like GMRES, without the need to devise and compute a preconditioner. Specifically, for simulations of beads interacting hydrodynamically with commonly encountered holonomic constraints, PrCG exhibits linear scaling with the number of beads similar to GMRES. However, because PrCG does not require a preconditioner it often requires less precomputational effort both in terms of computational complexity and storage, depending on the nature of the constraints. Furthermore, PrCG produces iterates which are always feasible, allowing for independent tuning of the error tolerance on the bead velocities from the error tolerance on the constraint equations. We demonstrate PrCG's efficiency by solving several example low Reynolds number systems involving rigid bodies, freely jointed chains, and immobile particles, as well as a mixed constraint problem with both a rigid body and immobile particles, comparing to GMRES and analytical solutions, when available.